How do you flirt with shy people?

If you think the person is shy, give them more time to show them that she likes them. You probably already know to not be pushy, but I also advise you not to give up so fast if you know they are reserved. Start with small steps that are low stakes like asking the person on a coffee date and then progress to “official” dates, like movies and dinner. Ask to hold their hand and don't lunge in for a kiss on the first date, or even the first few dates if you sense they are reserved. If they have been agreeing to go out on low key dates and holding your hand, then ask for more intimate dates and ask if you can kiss them.